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On Bayesian supremum norm contraction rates. (English) Zbl 1305.62189
Summary: Building on ideas from [I. Castillo and R. Nickl, Ann. Stat. 41, No. 4, 1999–2028 (2013; Zbl 1285.62052)], a method is provided to study nonparametric Bayesian posterior convergence rates when “strong” measures of distances, such as the sup-norm, are considered. In particular, we show that likelihood methods can achieve optimal minimax sup-norm rates in density estimation on the unit interval. The introduced methodology is used to prove that commonly used families of prior distributions on densities, namely log-density priors and dyadic random density histograms, can indeed achieve optimal sup-norm rates of convergence. New results are also derived in the Gaussian white noise model as a further illustration of the presented techniques.

62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62G05 Nonparametric estimation
62G07 Density estimation
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